Gimel function

<h2 id="values-of-the-gimel-function">Values of the gimel function</h2>
The gimel function has the property 
 
 
 
 ℷ
 (
 κ
 )
 >
 κ
 
 
 {\displaystyle \gimel (\kappa )>\kappa }
 
 for all infinite cardinals 
 
 
 
 κ
 
 
 {\displaystyle \kappa }
 
 by <a href="/facts/K%25C3%25B6nig%2527s_theorem_(set_theory)/fUaNe2PK">König's theorem</a>.
For regular cardinals

κ
      
    
    {\displaystyle \kappa }
  
,

ℷ
 (
 κ
 )
 =
 
 2
 
 κ
 
 
 
 
 {\displaystyle \gimel (\kappa )=2^{\kappa }}
 
, and <a href="/facts/Easton%2527s_theorem/g6o8sRrU">Easton's theorem</a> says we don't know much about the values of this function. For singular

κ
 
 
 {\displaystyle \kappa }
 
, upper bounds for 
 
 
 
 ℷ
 (
 κ
 )
 
 
 {\displaystyle \gimel (\kappa )}
 
 can be found from <a href="/facts/Saharon_Shelah/oI5WSK8p">Shelah</a>'s <a href="/facts/PCF_theory/o8FmPmRr">PCF theory</a>.

<h2 id="the-gimel-hypothesis">The gimel hypothesis</h2>
The gimel hypothesis states that 
 
 
 
 ℷ
 (
 κ
 )
 =
 max
 (
 
 2
 
 
 cf
 
 (
 κ
 )
 
 
 ,
 
 κ
 
 +
 
 
 )
 
 
 {\displaystyle \gimel (\kappa )=\max(2^{{\text{cf}}(\kappa )},\kappa ^{+})}
 
. In essence, this means that 
 
 
 
 ℷ
 (
 κ
 )
 
 
 {\displaystyle \gimel (\kappa )}
 
 for singular 
 
 
 
 κ
 
 
 {\displaystyle \kappa }
 
 is the smallest value allowed by the axioms of <a href="/facts/Zermelo%25E2%2580%2593Fraenkel_set_theory/UYIsPbXw">Zermelo–Fraenkel set theory</a> (assuming consistency).
Under this hypothesis cardinal exponentiation is simplified, though not to the extent of the <a href="/facts/Continuum_hypothesis/R0yGqHND">continuum hypothesis</a> (which implies the gimel hypothesis).

<h2 id="reducing-the-exponentiation-function-to-the-gimel-function">Reducing the exponentiation function to the gimel function</h2>
Bukovský (1965) showed that all cardinal exponentiation is determined (recursively) by the gimel function as follows. 

<ul><li>If 
 
 
 
 κ
 
 
 {\displaystyle \kappa }
 
 is an infinite regular cardinal (in particular any infinite successor) then 
 
 
 
 
 2
 
 κ
 
 
 =
 ℷ
 (
 κ
 )
 
 
 {\displaystyle 2^{\kappa }=\gimel (\kappa )}
 
</li>
<li>If 
 
 
 
 κ
 
 
 {\displaystyle \kappa }
 
 is infinite and singular and the continuum function is eventually constant below 
 
 
 
 κ
 
 
 {\displaystyle \kappa }
 
 then 
 
 
 
 
 2
 
 κ
 
 
 =
 
 2
 
 <
 κ
 
 
 
 
 {\displaystyle 2^{\kappa }=2^{<\kappa }}
 
</li>
<li>If 
 
 
 
 κ
 
 
 {\displaystyle \kappa }
 
 is a limit and the continuum function is not eventually constant below 
 
 
 
 κ
 
 
 {\displaystyle \kappa }
 
 then 
 
 
 
 
 2
 
 κ
 
 
 =
 ℷ
 (
 
 2
 
 <
 κ
 
 
 )
 
 
 {\displaystyle 2^{\kappa }=\gimel (2^{<\kappa })}
 
</li></ul>
The remaining rules hold whenever 
 
 
 
 κ
 
 
 {\displaystyle \kappa }
 
 and 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
 are both infinite:

<ul><li>If ℵ0 ≤ κ ≤ λ then κλ = 2λ</li>
<li>If μλ ≥ κ for some μ < κ then κλ = μλ</li>
<li>If κ > λ and μλ < κ for all μ < κ and cf(κ) ≤ λ then κλ = κcf(κ)</li>
<li>If κ > λ and μλ < κ for all μ < κ and cf(κ) > λ then κλ = κ</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Aleph_number/kwe9I4v5">Aleph number</a></li>
<li><a href="/facts/Beth_number/zNRDWG5Q">Beth number</a></li></ul>

<ul><li>Bukovský, L. (1965), "The continuum problem and powers of alephs", Comment. Math. Univ. Carolinae, 6: 181–197, <a href="/facts/Hdl_(identifier)/rdebSxmC">hdl</a>:<a href="https://hdl.handle.net/10338.dmlcz%2F105009">10338.dmlcz/105009</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0183649">0183649</a></li>
<li>Jech, Thomas J. (1973), <a href="http://matwbn.icm.edu.pl/ksiazki/fm/fm81/fm8116.pdf">"Properties of the gimel function and a classification of singular cardinals"</a> (PDF), Fund. Math., Collection of articles dedicated to Andrzej Mostowski on the occasion of his sixtieth birthday, I., 81 (1): 57–64, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.4064%2Ffm-81-1-57-64">10.4064/fm-81-1-57-64</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0389593">0389593</a></li>
<li><a href="/facts/Thomas_Jech/E4JQu4Mk">Thomas Jech</a>, Set Theory, 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-44085-2.</li></ul>

Gimel function open-in-new

Gimel function